Numerical calculation of the reflectance of sub-wavelength structures

Computer Physics Communications 180 (2009) 1721–1729

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Computer Physics Communications www.elsevier.com/locate/cpc

Numerical calculation of the reflectance of sub-wavelength structures on silicon nitride for solar cell application Kartika Chandra Sahoo a , Yiming Li b,c,∗ , Edward Yi Chang a a b c

Department of Materials Science and Engineering, National Chiao Tung University, Hsinchu 300, Taiwan Department of Electrical Engineering, National Chiao Tung University, Hsinchu 300, Taiwan National Nano Device Laboratories, Hsinchu 300, Taiwan

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 2 March 2009 Received in revised form 9 April 2009 Accepted 19 April 2009 Available online 22 April 2009 Keywords: Silicon nitride Sub-wavelength structure Pyramid shape Antireflection coating Multilayer Rigorous coupled-wave approach Reflectance Efficiency Morphological effect

In this study, we calculate the spectral reflectivity of pyramid-shaped silicon nitride (Si3 N4 ) sub-wavelength structures (SWS). A multilayer rigorous coupled-wave approach is advanced to investigate the reflection properties of Si3 N4 SWS. We examine the simulation results for single layer antireflection (SLAR) and double layer antireflection (DLAR) coatings with SWS on Si3 N4 surface, taking into account effective reflectivity over a range of wavelengths and solar efficiency. The results of our study show that a lowest effective reflectivity of 1.77% can be obtained for the examined Si3 N4 SWS with the height of etched part of Si3 N4 and the thickness of non-etched layer of 150 and 70 nm, respectively, which is less than the results of an optimized 80 nm Si3 N4 SLAR (∼5.41%) and of an optimized DLAR with 80 nm Si3 N4 and 100 nm magnesium fluoride (∼5.39%). 1% cell efficiency increase is observed for the optimized Si solar cell with Si3 N4 SWS, compared with the cell with single layer Si3 N4 antireflection coatings (ARCs); furthermore, compared with DLAR coated solar cell, the increase is about 0.71%. The improvement on the cell efficiency is mainly due to lower reflectance of Si3 N4 SWS over a wavelength region from 400 to 600 nm that leads to lower short circuit current. © 2009 Elsevier B.V. All rights reserved.

1. Introduction The antireflection coating is a key factor for solar cell design [1–3]. Many studies have been reported for double layer antireflection (DLAR) coatings because single layer antireflection (SLAR) coatings are not able to cover a broad range of the solar spectrum [4–8]. Unfortunately, these multilayer antireflection coatings (ARCs) are expensive to fabricate owing to the stringent requirement of high vacuum, material selection, and layer thickness control. An alternative to multilayer ARCs is the sub-wavelength structures (SWS) surface with dimensions smaller than the wavelength of light. In publications concerning broadband or solar antireflection surfaces [9–12], the principle to achieve the necessary low refractive indices is always the same: substrate material is mixed with air on a sub-wavelength scale. To date, a wide variety of techniques were examined for texturing microcrystalline-Si cells [13– 16]. One of promising options is surface texturing by dry etching technique. Fabricating uniform textures with a submicron scale on mc-Si wafers by reactive ion etching (RIE) for Si solar cells [17,18] has been studied. But, this may form the dislocations and defects

*

Corresponding author at: Department of Electrical Engineering, National Chiao Tung University, Hsinchu 300, Taiwan. E-mail address: [email protected] (Y. Li). 0010-4655/$ – see front matter doi:10.1016/j.cpc.2009.04.013

©

2009 Elsevier B.V. All rights reserved.

in the semiconductor layer. These defects and dislocations are responsible for increasing the minority carrier recombination in solar cell. Thus, the short circuit current for the solar cell is decreased, which in turn decreases the efficiency of solar cell. Also, the reflectance property of textured antireflection coatings has not been clearly drawn yet for Si solar cells. Based upon this observation, study the possibility of sub-wavelength structure on ARC surface instead of semiconductor surface may benefit the Si solar cell technologies. In this work, we numerically study the reflectance of SWS on silicon nitride (Si3 N4 ) for solar cell application. Since Si3 N4 is a well-known ARC used in semiconductor industry, we explore the texturization on Si3 N4 ARC and its optical properties. The main motivation behind this lies in the fact that the sub-wavelength structures will act as a second ARC layer with an effective refractive index so that the total structure can perform as a DLAR layer. Thus, we could cost down the deposition of 2nd ARCs layer can be saved with better or comparable performance as that of a DLAR solar cell. We calculate the spectral reflectivity of pyramidshaped Si3 N4 SWS. A multilayer rigorous coupled-wave approach (RCWA) [19–21] is advanced to investigate the reflection properties of Si3 N4 SWS. In contrast to conventional constant refractive index expression for Si, a wavelength-dependent expression is implemented in our calculation of reflectance of Si3 N4 SWS. We first optimize structures including Si SWS, Si3 N4 SWS, Si3 N4 SLAR and

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Si3 N4 /magnesium fluoride (MgF2 ) DLAR for the best effective reflectance properties. Using the optimized morphologies, we further compare the simulation results of reflectance for SLAR and DLAR coatings with SWS on Si3 N4 , over a range of wavelengths. The reflectance data obtained from RCWA simulation for the optimized SWS is used in PC1D [22] to get the electrical data (e.g., short circuit current J SC and open circuit voltage V OC ) and cell efficiency; we notice that PC1D is one of the commonly used software for solar cell modeling. The solar efficiencies obtained from PC1D simulation for SWS, SLAR coating and DLAR coating are also compared and discussed. This paper is organized as follows. In Section 2, we brief the procedure of RCWA for the simulated single pyramidal structure including the adopted material parameters. In Section 3, we show

the results and compare the main difference of reflectance among structures. Finally, we draw conclusions and suggest future work. 2. Structure and solution method For the simplicity, a single pyramidal structure, shown in Fig. 1(a), is explored for the reflectance property with respect to the wavelength. The region with brown color of SWS is Si3 N4 , the region with sky color stands for Si substrate, and the environment of the triangular part is air. The etched Si3 N4 (i.e., the height of triangular part) is h and the thickness of the non-etched Si3 N4 is s, both of these two parameters are designing parameters for the reflectance optimization. The studied SWS is a diffractive structure and its reflectance property could be calculated by a rigorous coupled-wave analysis (RCWA) technique. RCWA is an exact

Fig. 1. (a) Geometry of sub-wavelength structure studied in this work, where h is the height and s is the non-etched part of SWS. (b) A stack of uniform homogeneous layers resulting from the partitioned geometry of (a) for the reflection calculation using the multilayer RCWA method and EMT. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

K.C. Sahoo et al. / Computer Physics Communications 180 (2009) 1721–1729

solution of Maxwell’s equations for the electromagnetic diffraction by grating structures. A multilayer RCWA method is used in this study, where the effective medium theory (EMT) [23–25] is adopted to calculate the effective refractive index for each partitioned uniform homogeneous layer, as shown in Fig. 1(b). Without loss of generality (WLOG), we first divide the pyramidal structure into several horizontal layers with equal thickness. As shown in Fig. 1(b), for each discrete position zl along the zdirection, EMT implies that the effective refractive index n( zl ) of each layer is approximated by

   [1 − f (zl ) + n2 ][ f (zl ) + (1 − f (zl ))n2 ] + n2 SiN SiN SiN , n( zl ) =  (1) 2[ f ( zl ) + (1 − f ( zl ))n2SiN ] √ where f ( zl ) = 4rl2 / 3D 2 is the fraction of Si3 N4 contained in each

layer, rl is the base width for each layer and D is the base width of the structure, nSiN = n + ik is the complex refractive index of √ Si3 N4 , i = −1, n and k are optical constants, and nair = 1 is the refractive index of air. Note that only the real part of refractive index of Si3 N4 is considered in our simulation because it is weakly absorbing material [26]. With the calculated effective refractive index n( zl ) for each layer, we can solve the reflectance property of the entire structure including a layer for the non-etched Si3 N4 with respect to the different wavelength. From the partitioned structure, shown in Fig. 1(b), we now consider the reflection and the transmission of a transverse electric (TE) polarized plane wave of free-space wavelength λ, incident at angle θ , on L uniform layers of effective refractive indices n1 = n( z1 ), . . . , nl , . . . , n L = n( z L ) and thickness d1 , . . . , dl , . . . , d L . For each layer, the normalized electric field (in the x– y plane) for the input and the output regions is given by, for the air region, i.e., for z  0





E 0 = e −ikair ,z z + R × e −ikair ,z z e −ikx x ,

(2)

for the region of SWS, i.e., for D l−1  z  D l ,





E l = pl × e −ik0 γl (z− D l−1 ) + Q l × e ik0 γl (z− D l ) e −ikx x ,

(3)

and for the Si substrate, i.e., for z  D L , E t = T × e −i (kx x+k Si,z (z− D L )) ,

(4)

where kx = k0 nair sin θ,

(5)

kair,z = k0 nair cos θ,

(6)

kSi,z = k0 n2Si − n2air sin2 θ,

(7)

γl = i nl2 − n2air sin2 θ,

(8)





Dl =

l 

dp ,

(9)

p =1

l = 1, . . . , L, I , R and T are the incident, reflected and the transmitted amplitudes of the electric fields, P and Q are the field amplitudes in the uniform Si3 N4 slab, k0 = 2π /λ is the wavevector magnitude, and nair and nSi are the refractive indices of the air and the silicon regions. Note that now the layer of nonetched Si3 N4 has been added into our simulation structure, where its effective refractive index nSiN = 2.05 is the same with the original one and f ( z L ) = 1. The reflected and transmitted amplitudes of the explored SWS are calculated by matching the tangential electric- and magnetic-field components at the boundaries among layers [27]. First, for the boundary between the input air region and the first layer of Si3 N4 (i.e., z = 0), we have



1 + R = P 1 + Q 1 × e −k0 γ1 d1 , k ,z i air (1 − k0

R ) = γ1 ( P 1 − Q 1 × e −k0 γ1 d1 ).

(10)

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For the boundary between the (l − 1)st and the lth layers (i.e., z = D l−1 )



P l−1 × e −k0 γl−1 dl−1 + Q l−1 = P l + Q l × e −k0 γl dl ,

γl−1 ( P l−1 × e −k0 γl−1 dl−1 − Q l−1 ) = γl ( P l − Q l × e −k0 γl dl );

(11)

for the boundary between the last layer and the output Si region (i.e., z = D L ), the matched equations are



P L × e −k0 γL d L + Q L = T ,

γL ( P L × e −k0 γL d L − Q L ) = i (

(12)

kSi,z )T . k0

Eqs. (10)–(12) could be solved by using a transmittance matrix method [28]. Using Eq. (12), the field amplitudes P L and Q L in terms of the transmitted coefficient T are determined firstly. They are then substituted into Eq. (11) for the field amplitudes P L −1 and Q L −1 . Consequently, the system of equations to be solved for the reflection properties is given by





1 i



+

kair,z k0 L 

=

l=1

−i

kair,z k0

R



e −k0 γl dl

1

γl −γl e −k0 γl dl



×



1

e −k0 γl dl

1

 −1

γl e −k0 γl dl −γl

 ×

1 i

kSi,z k0

T,

(13)

for partitioned layers of SWS. Similarly, a set of governing equations could be derived for the transverse magnetic (TM) polarization. Here the incident angle θ of sun light is assumed to be normal to the plane (i.e., θ = 0◦ ), and only the TE polarization is considered here for the calculation of the reflection properties [29]. For a given number of layers for the SWS including the layer of non-etched Si3 N4 , say L in total; a calculation procedure for computing the reflectance properties of the studied SWS described above is summarized: (i) calculate the effective refractive index for each zl via Eq. (1); (ii) compute the coefficients using Eqs. (5)–(9) for a specified wavelength λ; and (iii) solve Eq. (13) to get the unknowns R and T . The reflectance spectra obtained from RCWA simulation above are used in PC1D simulation to compare the effect on the short circuit current density ( J SC ), open circuit voltage (V OC ) and efficiency (η ) for a solar cell structure. 3. Results and discussion First of all, we compare the reflectance with respect to the wavelength of sunlight for the SWS with Si and Si3 N4 . As shown in Fig. 2, by assuming a constant refractive index for Si nSi = 3.875 and for Si3 N4 nSiN = 2.05, the reflectance versus the wavelength for Si SWS with h = 88 nm and Si3 N4 SWS with h = 88 nm and vanished non-etched part of SWS (i.e., s = 0 nm) is simulated and compared. It is found that the non-optimized Si3 N4 SWS possesses a little bit higher reflectance which may not be a plus for ARC. However, we can design a Si3 N4 SWS with the case of non-zero s, say h = 68 nm and s = 20 nm, which shows that the reflectance is close to the result of Si SWS or even better. This observation motivates us to explore the morphology-dependent reflectance of Si3 N4 SWS with a set of optimized h and s. Note that Si refractive index nSi may depend upon the wavelength of incident sunlight [30]; our calculation for the Si3 N4 SWS with h = 68 nm and s = 20 nm confirms the reflectance difference between the model with constant and wavelength-dependent nSi , as shown in Fig. 3. In this calculation, an empirically fitted formula for the wavelength-dependent nSi is implemented in our simulation program [31]



nSi =

ε+

A

λ2

+

B λ21

(λ2 − λ21 )

,

(14)

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Fig. 2. The reflectance versus the wavelength for Si SWS with h = 88 nm and Si3 N4 SWS with h = 88 nm and vanished non-etched part of SWS (i.e., s = 0 nm). Comparison for the Si3 N4 SWS with the case of non-zero s, say s = 20 nm is also provided. It is found that it is possible to reduce the reflectance of Si3 N4 SWS by proper selection of h and s.

Fig. 3. The spectral reflectivity of Si3 N4 SWS for h = 68 nm and s = 20 nm with constant Si refractive index and refractive index as a function of lambda given by Eq. (14).

where, λ1 = 1.1071 μm, ε = 11.6858, A = 0.939816 and B = 8.10461 × 10−3 . Instead of considering the reflectance for a certain wavelength, an effective reflectance is further computed for the structures over a range of the wavelength of incident sunlight. By taking purely Si in the SWS part of Fig. 1(a), where s = 0 nm and h is designed as a varying factor, we now calculate the effective reflectance R eff [32] for the wavelength λ varying from λl = 400 nm to λu = 1000 nm and compare it with Si3 N4 SWS. R eff is evaluated by

 λu R eff =

R (λ)SI(λ) dλ E (λ) ,  λu SI(λ) λl E (λ) dλ

λl

(15)

where, SI(λ) is spectral irradiance given by ATMG173 AM1.5G reference [33], E (λ) is photon energy and R (λ) is the calculated reflection. Figs. 4(a) and 4(b) show the effective reflectance as a function of h for Si SWS, and of h and s for Si3 N4 SWS. For Si SWS, there is a minimum R eff = 3.89% for h = 220 nm, and for Si3 N4 SWS, the minimum of R eff = 1.77% occurs at h = 150 nm and s = 70 nm. Compared with Si SWS, the twice improvement of R eff for Si3 N4 SWS is due to the nature of Si3 N4 and an optimal combination of the height of etched part of Si3 N4 and the thickness of non-etched part of Si3 N4 . It has been reported that SLAR and DLAR were used in solar cell, for a unified comparison, similarly, we further examine their R eff over the same wavelength, as shown in Figs. 5(a) and 5(b).

K.C. Sahoo et al. / Computer Physics Communications 180 (2009) 1721–1729

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Fig. 4. The effective reflectance for the wavelength varying from 400 to 1000 nm; plot is as a function of (a) h for Si SWS and (b) of h and s for Si3 N4 SWS.

For Si3 N4 SLAR coating on Si, as shown in the inset of Fig. 5(a), the refractive index is equal to 2.05, where the thickness of ARC is varied. For Si3 N4 /ARC 2 DLAR coating on Si, the thickness of ARC 2 and the refractive index of ARC 2 are varied. Note that the thickness of ARC 1 equals 80 nm, as shown in the inset of Fig. 5(b), directly comes from the optimal value of Fig. 5(a), and the lower bound of refractive index of ARC 2 starts from 1.38 which is the refractive index of MgF2 . The lowest R eff occurs

when the refractive index of ARC 2 is 1.38 and its thickness is 100 nm. Based upon the investigation of Figs. 4 and 5, we show the optimal reflectance spectra among the bulk Si (i.e., the bare Si), the optimized SLAR, DLAR, Si SWS and Si3 N4 SWS for the wavelength from 400 to 1000 nm in Fig. 6. Table 1 lists the effective reflectivity for those optimized structures of 150 nm Si3 N4 SWS and 70 nm non-textured Si3 N4 film, com-

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Fig. 5. Plot of the effective reflectance for the wavelength varying from 400 to 1000 nm. (a) is the result as a function of the thickness of Si3 N4 ARC with n = 2.05 for SLAR coating on Si. (b) is the result as a function of the thickness of ARC 2 and refractive index of ARC 2 for Si3 N4 /ARC2 DLAR coating on Si. The thickness of Si3 N4 ARC 1 is fixed at 80 nm which is optimized from (a).

pared with the Si SWS, Si3 N4 SLAR (its thickness is 80 nm) and Si3 N4 /MgF2 DLAR (its thickness is 80–100 nm) structures. The flat silicon substrate exhibits high reflection >35% for visible and near infrared wavelengths, Si3 N4 SLAR coatings exhibits low reflection 35% for shorter wavelengths 400 nm, and Si3 N4 /MgF2 DLAR coatings exhibits low reflection 20% for short wavelength 400 nm, while the SWS gratings show reduced reflection of